We present the convergence of the proposed algorithm by plotting the wsr against the number of iterations in Fig. REF . The simulation parameters and the network topology are same as those used in REF , and we fix the distance \(L\)  as \(L=25\,{\rm m}\) . To show the convergence in non-ideal low-resolution phase shift cases, we add the curves “1-bit phase shift” and “2-bit phase shift” to denote the case \({\cal F}={\cal F}_3\) , where the discrete available phase shift sets are \(\lbrace 0,\pi \rbrace \)  and \(\lbrace 0,\pi /2,\pi ,3\pi /2\rbrace \) , respectively. The results in Fig. REF  clearly illustrate that, when the convergence error is no more than \(2\%\) , the proposed joint precoding framework can converge within 15 iterations. To be more specific, the ideal ris case and continuous phase shift case can converge within 15 iterations. The discrete phase shift case can converge within 10 iterations. Since the conventional cell-free network without RIS does not need to address the passive precoding design at RISs, the WSR in the case “No RIS” converges within 8 iterations. The results indicate that, although the approximation operation in (REF ) causes the uncertainty of convergence, the proposed framework still enjoys the global convergence and a fast convergence speed. Besides, we can also notice that, the low-resolution phase shifts only suffer from an acceptable performance loss. For example, after 15 iterations, the wsr in “Ideal RIS case” is about \(82\,{\rm bit/s/Hz}\) , while that of the cases “2-bit phase shift” is about \(75\,{\rm bit/s/Hz}\) , which suffers a loss about \(8\%\)  of the ideal case. Considering the hardware implementation of RIS [1]}, since the low-resolution phase shift is much easier for implementation than the continuous one, this result is very encouraging for the massive deployment of RIS in the RIS-aided cell-free network.
